Calculus of Variations and Geometric Measure Theory

S. Dipierro - A. Pinamonti - E. Valdinoci

Rigidity results for elliptic boundary value problems: stable solutions for quasilinear equations with Neumann or Robin boundary conditions

created by pinamonti on 22 Sep 2017
modified on 14 Mar 2018


Accepted Paper

Inserted: 22 sep 2017
Last Updated: 14 mar 2018

Journal: IMRN
Year: 2017


We provide a general approach to the classification results of stable solutions of (possibly nonlinear) elliptic problems with Robin conditions.

The method is based on a geometric formula of Poincar\'e type, which is inspired by a classical work of Sternberg and Zumbrun and which gives an accurate description of the curvatures of the level sets of the stable solutions. {F}rom this, we show that the stable solutions of a quasilinear problem with Neumann data are necessarily constant.

As a byproduct of this, we obtain an alternative proof of a celebrated result of Casten and Holland, and Matano.

In addition, we will obtain as a consequence a new proof of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.